Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Resolvent estimates for elliptic finite element operators in one dimension


Authors: M. Crouzeix, S. Larsson and V. Thomée
Journal: Math. Comp. 63 (1994), 121-140
MSC: Primary 65N30; Secondary 65M60
DOI: https://doi.org/10.1090/S0025-5718-1994-1242058-1
MathSciNet review: 1242058
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the analyticity (uniform in h) of the semigroups generated on $ {L_p}(0,1),1 \leq p \leq \infty $, by finite element analogues $ {A_h}$ of a one-dimensional second-order elliptic operator A under Dirichlet boundary conditions. This is accomplished by showing the appropriate estimates for the resolvents by means of energy arguments. The results are applied to prove stability and optimal-order error bounds for numerical solutions of the associated parabolic problem for both smooth and nonsmooth data.


References [Enhancements On Off] (What's this?)

  • [1] G. A. Baker, J. H. Bramble, and V. Thomée, Single step Galerkin approximations for parabolic problems, Math. Comp. 31 (1977), 818-847. MR 0448947 (56:7252)
  • [2] P. Brenner, M. Crouzeix, and V. Thomée, Single step methods for inhomogeneous linear differential equations in Banach space, RAIRO Anal. Numér. 16 (1982), 5-26. MR 648742 (83d:65268)
  • [3] M. Crouzeix, S. Larsson, S. Piskarev, and V. Thomée, The stability of rational approximations of analytic semigroups, BIT 33 (1993), 74-84. MR 1326004 (96f:65069)
  • [4] M. Crouzeix and V. Thomée, The stability in $ {L_p}$ and $ W_p^1$ of the $ {L_2}$-projection onto finite element function spaces, Math. Comp. 48 (1987), 521-532. MR 878688 (88f:41016)
  • [5] G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. 54 (1975), 305-387. MR 0442749 (56:1129)
  • [6] G. Da Prato and E. Sinestrari, Differential operators with non dense domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 285-344. MR 939631 (89f:47062)
  • [7] J. Douglas, Jr., T. Dupont, and L. Wahlbin, Optimal $ {L_\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29 (1975), 475-483. MR 0371077 (51:7298)
  • [8] S. Larsson, V. Thomée, and L. B. Wahlbin, Finite-element methods for a strongly damped wave equation, IMA J. Numer. Anal. 11 (1991), 115-142. MR 1089551 (92d:65164)
  • [9] B. S. Mitjagin and E. M. Semenov, Absence of interpolation of linear operators in spaces of smooth functions, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 1289-1328; English transl. in Math. USSR Izv. 11 (1977), 1229-1266. MR 0482148 (58:2234)
  • [10] C. Palencia, A stability result for sectorial operators in Banach spaces, SIAM J. Numer. Anal. 30 (1993), 1373-1384. MR 1239826 (94j:65109)
  • [11] -, On the stability of variable stepsize rational approximations of holomorphic semigroups, Math. Comp. 62 (1994), 93-103. MR 1201070 (94c:47066)
  • [12] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, 1983. MR 710486 (85g:47061)
  • [13] E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl. 107 (1985), 16-66. MR 786012 (86g:34086)
  • [14] V. Thomée and L. B. Wahlbin, Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable, Numer. Math. 41 (1983), 345-371. MR 712117 (85f:65099)
  • [15] L. B. Wahlbin, A quasioptimal estimate in piecewise polynomial Galerkin approximation of parabolic problems, Numerical methods, Proceedings, Dundee 1981 (G. A. Watson, ed.), Lecture Notes in Math., vol. 912, Springer-Verlag, 1981, pp. 230-245. MR 654353 (83f:65157)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 65M60

Retrieve articles in all journals with MSC: 65N30, 65M60


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1994-1242058-1
Keywords: Resolvent, sectorial, elliptic, analytic semigroup, Banach space, $ {L_p}$, maximum norm, finite element method, rational approximation, stability, error estimate
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society